Bond Prices and Yields
BUSI 721, Fall 2022
JGSB, Rice University
Kerry Back
Coupons vs Yields
The coupon rate of a bond is set at the time of its issue.
However, what one anticipates earning on a bond varies with the market price.
- Price < par \(\Rightarrow\) coupon + capital gain
- Price > par \(\Rightarrow\) coupon - capital loss
What one would earn per year on a bond if held to maturity (assuming no default) is called the bond yield.
Definition of bond yield
- Buy a bond at price \(P\), receive coupons \(c_1, \ldots, c_n\) and face value \(F\) at date \(n\).
- The rate of return \(r\) for which an investment of \(P\) would exactly finance the cash flows \(c_1, \ldots, c_n\) and \(F\) is called the bond yield.
- It is the rate at which the present value of the promised cash flows \(c_1, \ldots, c_n\) and \(F\) equals the bond price.
Calculating Yields
- Bonds usually pay coupons semi-annually, so it is conventional to use semi-annual discounting.
- Calculate a six-month rate. Double it to annualize.
- Suppose the next coupon payment is six months away. Solve the following for a six-month rate \(r\). Yield is \(y=2r\).
\[P=\frac{c}{1+r}+\frac{c}{(1+r)^2}+\cdots+\frac{c+F}{(1+r)^n}\]
Example
5 year bond, 6% coupon, $100 face
\[P=\frac{3}{1+r}+\frac{3}{(1+r)^2}+\cdots+\frac{3+100}{(1+r)^{10}}\]
5 year bond, 6% coupon, $100 face
Price sensitivity to market rates
- When market rates rise, a bond price falls until its yield is commensurate with market rates.
- When market rates fall, a bond price rises until its yield is commensurate with market rates.
So,
- rising rates \(\Rightarrow\) negative bond returns
- falling rates \(\Rightarrow\) positive bond returns
History of Treasury rates
Credit ratings
- Issuers are rated by S&P and Moody’s (and Fitch)
- AAA=best quality
- AA
- A
- BBB=worst investment grade
- BB=best non-investment grade (= junk = high-yield)
- B
- etc.
- There are +’s and -’s. Moody’s uses baa instead of BBB, etc.
Credit spreads
- Investors require higher yields on riskier bonds
- default \(\rightarrow\) low return
- no-default \(\rightarrow\) high return (= yield)
- yield must be high enough so that expected return = Treasury yield + risk premium
- Usually think of yield as Treasury yield + spread
- Part of spread offsets expected loss due to default
- Remainder provides a risk premium
History of credit spreads